I have been trying to solve this problem for hours now, but can write the solution down.
Intuitively $U$ will be distribued $R(0,2)$ since every real number $]0,1[$ $[1,2[$ has equal change by the fact $Z \sim b(1, 1/2)$

Thanks, never tried LaTeX but I will look into it. The distribution class $R(0,1)$ it the real numbers $]0,1[$ It just means that every real number in this interval is mapped to $density = 1$ using the mass-density function $f_X(x)$
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Nicolas Lykke IversenJan 15 '13 at 12:14